First derivative of the hard-sphere radial distribution function at contact

Analysis of probability of inserting a hard spherical particle with small diameter in hard-sphere fluid

The probability of inserting, without overlap, a hard spherical particle of diameter σ in a hard-sphere fluid of diameter σ0 and packing fraction η determines its excess chemical potential at infinite dilution, μex(σ, η). In our previous work [R. L. Davidchack and B. B. Laird, J. Chem. Phys. 157, 074701 (2022)], we used Widom's particle insertion method within molecular dynamics simulations to obtain high precision results for μex(σ, η) with σ/σ0 ≤ 4 and η ≤ 0.5. In the current work, we investigate the behavior of this quantity at small σ. In particular, using the inclusion-exclusion principle, we relate the insertion probability to the hard-sphere fluid distribution functions and thus derive the higher-order terms in the Taylor expansion of μex(σ, η) at σ = 0. We also use direct evaluation of the excluded volume for pairs and triplets of hard spheres to obtain simulation results for μex(σ, η) at σ/σ0 ≤ 0.2247 that are of much higher precision than those obtained earlier with Widom's method. These results allow us to improve the quality of the small-σ correction in the empirical expression for μex(σ, η) presented in our previous work.

Approximate analytical structure and thermodynamics for penetrable sphere fluids using effective hard spheres

A penetrable sphere fluid is modeled using the effective hard sphere fluid and its known structure and thermodynamics. It is observed that the obtained equation of state and the radial distribution function are in fairly good agreement with computer simulation data both at low temperatures and/or low densities.

Monte Carlo simulations of fluids whose particles interact with a logarithmic potential

Monte Carlo simulations of a model fluid in which the particles interact via a continuous potential that has a logarithmic divergence at a pair separation of sigma, which we introduced in J. G. Powles et al., Proc. R. Soc. London, Ser. A 455, 3725 (1999), have been carried out. The potential has the form, phi(r)= -epsilon ln(fr), where epsilon sets the energy scale and fr=1-(sigma/r)m. The value of m chosen was 12 but the qualitative trends depend only weakly on the value of m, providing it is greater than 3. The potential is entirely repulsive and has a logarithmic divergence as approximately -ln(r/sigma-1) in the r-->sigma limit. Predictions of the previous paper that the internal energy can be computed at all temperatures using the standard statistical mechanics formula for continuous potentials are verified here. The pressure can be calculated using the usual virial expression for continuous potentials, although there are practical limitations in resolving the increasingly important contribution from the r-->sigma limit at reduced temperatures greater than approximately 5. The mean square force F2 and infinite frequency shear Ginfinity and bulk Kinfinity moduli are only finite for T*=kBT/epsilon<1. The logarithmic fluid's physical properties become increasingly more like that of the hard sphere fluid with increasing temperature, showing a sharp transition in the behavior of the mean square force and infinite frequency elastic constants at T*=1. The logarithmic fluid is shown to exhibit a solid-fluid phase transition.

Thermodynamic properties of inverse power fluids

The local scaling behavior of the radial distribution function of the soft sphere or inverse power, r-n potential, fluid leads to a formula for the equation of state. From this formula different analytic forms for the compressibility factor, Z, have been derived. In the first, Z is expressed as a product of three functions, the hard sphere equation of state and two other functions incorporating the effects of the potential softness. In the second formula, the compressibility factor is cast in terms of the position and height of the first peak in the radial distribution function. In the final form, Z can be expressed as an exponential function which depends entirely on a combination of the virial coefficients. In each case Z is an explicit expression which has the correct low density limiting behavior and is accurate up to the freezing density for all packing fractions and circa n>or=12. Expressions are derived for the various component functions required for the different forms of Z, and relations between them are established. The compressibility factor manifests a maximum value or "ridge" when plotted as contours on the density-softness plane. It starts for the softer fluids at lower densities, increases with particle stiffness, and crosses the freezing line at n congruent with 33. From the compressibility factor other thermodynamic quantities can be obtained and the density-softness dependence of the infinite frequency limit elastic properties been determined. A self-consistent expression is derived for the effective hard sphere packing fraction (or equivalently, diameter), valid for all packing fractions and circa n>12. The effective hard-sphere diameter is compared with the formulas of Barker and Henderson, and Wheatley.